Foundations — Cubic spline interpolation — natural, clamped

Before you can read the parent note, you must be able to read every mark on the page without flinching. Below is a from-zero tour of each symbol and concept, ordered so that each one leans only on the ones before it. If the parent note wrote it and assumed you knew it — it's built here.

1. Points, and the subscript trick

The mark $x_i$ is read "x-sub-i" and simply means "the $x$-value of point number $i$". So $x_0$ is the first, $x_1$ the next, up to $x_n$ (the last). The letter $n$ is the number of the last point, so there are $n+1$ points in total (because we started counting at $0$, not $1$).

Look at the figure: the dots are the $(x_i,y_i)$. The condition $x_0<x_1<\cdots <x_n$ (the parent writes exactly this) just says the points march strictly left to right — no two share an $x$, and none go backwards. This ordering is what lets us talk about "the interval between neighbours."

2. The interval and its width $h_i$

Picture: $h_i$ is the horizontal gap between two neighbouring dots (see figure s01, the orange bracket underneath). Why the topic needs it: every formula divides by $h_i$ — because a slope or a curvature is always "how much change per unit width," and you cannot measure change per width without knowing the width.

3. What a cubic is (and why cubic)

Picture: a single smooth wiggle that can go up then down (or vice-versa) — see figure s02. A straight line ($x^1$) can't bend; a parabola ($x^2$) bends one way only; a cubic can bend, straighten, and bend back — flexible enough to smoothly link two heights while also controlling slope and curvature at the joins.

4. Function notation $S(x)$ and the piecewise idea

So $S$ is piecewise: different formula on different stretches, but one continuous curve overall. The parent writes $S_i(x_i)=y_i$ — read that as "piece $i$, evaluated at the left node $x_i$, gives the data height $y_i$." That is the interpolation condition: the curve must pass exactly through the dots.

5. Slope: the first derivative $S^{\prime }$

Picture: the tilt of a tiny straight ruler laid tangent to the curve at a point (figure s03, the magenta tangent). Why we need it: if two pieces meet at a node but have different slopes, the curve has a visible kink — like the corner of a roof. Demanding $S_{i-1}^{\prime }(x_i)=S_i^{\prime }(x_i)$ (slopes agree from the left and the right) erases the kink. This is where the pieces "talk to each other."

6. Curvature: the second derivative $S^{\prime \prime }$

Picture: figure s03 shows two arcs — one gently curved (small $|S^{\prime \prime }|$) and one sharply curved (large $|S^{\prime \prime }|$). Why we need it: matching slope removes kinks in the tilt, but you can still have a subtle "bendiness jump." Matching $S^{\prime \prime }$ too makes the curve fair — the eye sees no abrupt change of curvature. Demanding $S$, $S^{\prime }$, and $S^{\prime \prime }$ all continuous is exactly what the parent calls being ==$C^2$==.

7. The moment $M_i$ — a nickname for end-curvature

Why work with $M_i$ at all? Because on a single interval, $S^{\prime \prime }$ is a straight line (a cubic's second derivative is linear). A straight line is completely pinned by its two endpoint values $M_i$ and $M_{i+1}$. So instead of chasing four coefficients per piece, we only track one number per node — far fewer unknowns, and they link up neatly into a solvable chain.

8. Slopes-of-slopes: the right-hand side is discrete curvature

The parent's master equation has a right side $6\left(\frac{y_{i+1}-y_i}{h_i}-\frac{y_i-y_{i-1}}{h_{i-1}}\right).$ Read the inner pieces: $\frac{y_{i+1}-y_i}{h_i}$ is "rise over run" = the average slope on the right interval; the other term is the average slope on the left interval. Their difference is how much the slope changed as you crossed node $i$ — a raw, data-only estimate of curvature. This "difference of differences" idea is the world of Finite differences. The $6$ is a constant that pops out of integrating a cubic twice; you'll take it on faith here.

9. Boundary conditions — the two missing facts

The parent counts $4n$ coefficient-dials but only $4n-2$ equations from interpolation + smoothness, leaving the system 2 short. Two more facts must be supplied at the two free ends:

Both plug in as the missing two rows. The word "natural" = No curvature; "clamped" = Controlled slope.

10. Tridiagonal system — the shape of the equations

Why care: a tridiagonal system is solved super-fast and stably by the Tridiagonal systems & Thomas algorithm — no need for heavy general matrix methods. This is a big practical reason splines are cheap to compute.

Prerequisite map

Read it downward: points give widths and cubic pieces; cubics give slope, then curvature, then moments; smoothness plus moments plus boundary facts assemble the tridiagonal system; solving it yields the spline.

Equipment checklist

Cover the answers and check you can produce each from memory before reading the parent.